Quantum information is physical information that is held in the state of a quantum system. The unit of quantum information may be a qubit, a two-level quantum system. In contrast to discrete classical digital states, a two-state quantum system can be in a superposition of the two states at any given time. Unlike classical information, quantum information cannot be read without the state being disturbed by the measurement device. Furthermore, in quantum information, an arbitrary state cannot be cloned.
Ising model is a mathematical model of ferromagnetism in statistical mechanics. The model comprises of discrete variables that represent magnetic dipole moments of atomic spins that can be in one of two states (+1 or −1). The spins are typically arranged in a graph, usually a lattice, allowing each spin to interact with its neighbors. The Ising model allows the identification of phase transitions, as a simplified model of reality. A two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition.
The adiabatic theorem is a concept in quantum mechanics which states that a quantum mechanical system prepared in a minimum-energy configuration and subjected to gradually changing external conditions adapts its state so as to always remain in an instantaneous energy-minimizing configuration subject to the external conditions, but when subjected to rapidly varying conditions there is insufficient time for the quantum system to adapt itself which may potentially result in the system making disrete transitions to high-energy states. Adiabatic quantum computation (AQC) relies on the adiabatic theorem to do calculations and is closely related to quantum annealing, which may be regarded as a subclass of AQC.
Quantum annealing is a powerful technique used to determine the solutions of constraint satisfaction problems through the quantum adiabatic theorem. The original proposal of quantum annealing uses a quantum Ising spin glass to encode the constraint satisfaction problems. This has been generalized to adiabatic quantum computation (AQC), which uses a wider class of initial and final Hamiltonians. The essential idea behind AQC is to encode the solution to the (usually, an optimization) problem into the ground state of a ‘final’ Hamiltonian HP, initialize a quantum system in the easy-to-prepare ground state of an ‘initial’ Hamiltonian H0, and tune the Hamiltonian slowly (adiabatically) over a time period T as, H(t)=(1−t/T) H0+(t/T) HP. If H0 and HP do not commute, and T is chosen large enough—i.e., T˜1/g3, where g is the minimum gap between the energy of the ground state and the first excited state of H(t) over the interval tin (0, T)—then the adiabatic theorem of quantum mechanics guarantees that the system will remain in the ground state of H(t) at all times t and hence the solution to the problem can be read out via a measurement of (the energy of) the system at t=T. AQC further led to the development of the adiabatic model of quantum computing, which was shown to be computationally as powerful as the more well-known circuit model of quantum computing.
There has been significant interest in AQC due to its potential ease of implementation. In particular, there has been progress towards building devices capable of implementing the aforesaid model for a restricted class of HP, for which H(t) corresponds to the Hamiltonian of the Ising model with a transverse magnetic field, sometimes known as the quantum Ising model. This restricted class of AQC is termed quantum annealing. The ground state of the quantum Ising model can encode the solutions of many NP hard optimization problems, and quantum annealing provably converges to the optimal solution of these problems as long as the conditions of the adiabatic theorem are satisfied. It is important to note however that the aforesaid restricted class of Hamiltonians belong to a special class called stoquastic Hamiltonians, which is insufficient to build a universal adiabatic quantum computer. Further, it is not known, for a general Ising model, whether quantum annealing provides any speedup, computational-complexity-wise, over the best-known classical algorithms.
In computational complexity theory, nondeterministic polynomial time (NP) is one of the most fundamental complexity classes. Intuitively, NP is the set of all decision problems for which the instances where the answer is “yes” have efficiently verifiable proofs of the fact that the answer is indeed “yes”. More specifically, these proofs have to be verifiable in polynomial time by a deterministic Turing machine. In an equivalent formal definition, NP is the set of decision problems where the “yes”-instances can be accepted in polynomial time by a non-deterministic Turing machine.
Even in the absence of a computational complexity advantage, an all-optical implementation of quantum annealing could arguably be quite attractive—both due to its potential ability to gain a large constant-factor speedup (due to optical modulation bandwidths that could far exceed the clock speeds of an electronic computer), as well as its potential ability to yield large power savings over classical electronic-computing solutions.
Quantum non-linear integrated photonics is a fast progressing field, and SOME recent developments have been made in realizing non-linear self and cross-phase modulation on photonic wave-guides and optical cavities. In a recent work, Gaeta and collaborators demonstrated the realization of cross-phase modulation of 10−3 radian per photon on a few-photon-bearing optical mode with ns-level response times. (See, cited reference [10] in the enclosed Appendix A, the entire contents of which is expressly incorporated by reference herein). They used rubidium vapor in a hollow-core photonic band gap fiber in a novel geometry that can tightly confine the optical mode over distances much greater than the diffraction length, in order to impart the non-linear phase shift. Further, recent work by Lukin and collaborators demonstrated a non-linear phase shift in a cavity quantum electrodynamics (QED) system, which realized an atom-photon interaction via multiple bounces of a single-photon-bearing mode in an optical cavity. (See, cited reference [11] in the enclosed Appendix A, the entire contents of which is expressly incorporated by reference herein). Finally, Englund and O'Brien, and their collaborators, have made recent advances in fabricating thermally-tuned fast-programmable linear-optical circuits. (See, cited references [12] and [13] in the enclosed Appendix A, the entire contents of which is expressly incorporated by reference herein). That is, nanophotonic circuits that can be programmed to realize arbitrary multi-spatial-mode passive unitary transformations, for example, ones that can be constructed via arbitrary configurations of beamsplitters and phase shifters.
Recently, some Ising models are implemented using injection locked lasers, where the Ising spins are mapped onto one of two (right or left) circular polarizations of an array of coupled slave laser oscillators that are driven by a strong master laser oscillator. In some other design, the Ising spins are mapped to a pair of orthogonal states of an array of coupled optical parametric oscillators. In both designs, the system is initialized in a random low-temperature state, and the Hamiltonian governing the Ising interactions, which is the Ising Hamiltonian without the transverse magnetic-field term is slowly turned on. Unlike in quantum annealing, during that turning-on phase, the system does not reside in the instantaneous Hamiltonian's ground state, but as the temperature is raised, is seen to converge to the energy-minimizing state, thereby obtaining the solution to the Ising problem. The most significant shortcoming of both of these approaches is that the fact that the state of the system always converges to the minimum-energy configuration for any instance of the Ising problem is not proven. In other words, the proposed optical Ising solvers do not converge to the optimal solution.
The approach in the disclosed invention exploits the theory of quantum annealing to propose the design of an Ising solver that provably converges to the optimal solution. Further, the proposed design can potentially be highly scalable to a large number of variables by exploiting recent advances in reconfigurable integrated photonic circuits.